Binomial Yields Research Articles

A heuristic solution method for disassemble-to-order problems with binomial disassembly yields

In disassemble-to-order problems, where a specific amount of several components must be obtained from the disassembly of several types of returned products, random disassembly yields create a formidable challenge for appropriate planning. In this context, it is typically assumed that yields from disassembly are either stochastically proportional or follow a binomial process. In the case of yield process misspecification, it has been shown (see Inderfurth et al. (2015)) that assuming binomial yields usually results in a lower penalty than assuming stochastically proportional yields. While there have been heuristics developed for the disassemble-to-order problem with stochastically proportional yields, a suitable, powerful heuristic for binomial yields is needed in order to facilitate solving problems with complex real-world product structures. We present a heuristic approach that is based on a decomposition procedure for the underlying non-linear stochastic optimization problem and that can be applied to problems of arbitrary size. A comprehensive numerical performance study using both randomly generated instances as well as a full factorial experimental design and, additionally, the data of a practical case example reveals that this heuristic delivers close-to-optimal results.

How yield process misspecification affects the solution of disassemble-to-order problems

Random yields from production are often present in manufacturing systems and there are several ways that this can be modeled. In disassembly planning for product recovery, e.g., we face high yield uncertainty in harvesting parts from cores. In this case, yield randomness often is modeled as either stochastically proportional or binomial. A statistical analysis of data from engine remanufacturing of a major car producer fails to provide conclusive evidence on which kind of yield randomness might prevail. In order to gain insight into the importance of this yield assumption, the impact of possible yield misspecification on the solution of the disassemble-to-order problem is investigated. Our results show that the penalty for misspecifying the yield method can be substantial, and provide insight on when the penalty would likely be problematic. The results also indicate that in the absence of conclusive information on which alternative should be chosen, presuming binomial yields generally leads to lower cost penalties and therefore preferable results.

Supplier Diversification under Inventory Holding Cost Increase and Binomial Yields

We consider supplier diversification with holding cost increasing based on binomial yields. We also show that the optimal solution always with a single supplier in the holding cost increasing binomial yields systems, and give the optimal solution.

On the supplier diversification under binomial yield

In a recent article, Fadiloglu et al. study an EOQ inventory model with multiple suppliers and binomial yields and conclude that sole sourcing is always optimal. This finding is contrary to the literature advocating diversification of the supply base in the presence of uncertainty. In this note, we present an alternative and more elegant proof of the optimality of sole sourcing in the EOQ model with binomial yield. Our approach extends this result to a general structure for the fixed order cost and provides more insights into the problem.

Multistage production systems with random yields and rigid demand

The increased complexity of contemporary manufacturing makes it hard to maintain a predictable level of output. We consider several types of multistage systems in an environment where yield is random, production is in lots and demand needs to be satisfied in its entirety. For each possible configuration of intermediate inventories a production policy must specify the stage at which to produce next and the lot size to be processed. The objective is to minimise the expected total of setup and variable production costs. We show that the expected cost associated with any arbitrary production policy can be calculated by solving a finite set of linear equations. The result is general in that it applies to any yield structure. We also develop efficient heuristics to solve various multistage systems. Selected numerical results are presented for binomial yields.

Supplier diversification under binomial yield

We consider supplier diversification in an EOQ type inventory setting with multiple suppliers and binomial yields. We characterize the optimal policy for the model and show that, in this case, it does not pay to diversify, in contrast to previous results in the random yield literature.

Lot-sizing two-echelon assembly systems with random yields and rigid demand

We consider a two-echelon assembly system producing a single final product for which the demand is known. The first echelon consists of several parallel stages, whereas the second echelon consists of a single assembly stage. We assume that the yield at each stage is random and that demand needs to be satisfied in its entirety; thus, several production runs may be required. A production policy should specify, for each possible configuration of intermediate inventories, on which stage to produce next and the lot size to be processed. The objective is to minimize the expected total of setup and variable production costs. We prove that the expected cost of any production policy can be calculated by solving a finite set of linear equations whose solution is unique. The result is general in that it applies to any yield distribution. We also develop efficient algorithms leading to heuristic solutions with high precision and, as an example, provide numerical results for binomial yields.

A two-bottleneck system with binomial yields and rigid demand

This study considers multistage production systems where production is in lots and only two stages have non-zero setup costs. Yields are binomial and demand, needing to be satisfied in its entirety, is “rigid”. We refer to a stage with non-zero setup cost as a “bottleneck” (BN) and thus to the system as “a two-bottleneck system” (2-BNS). A close examination of the simplest 2-BNS reveals that costs corresponding to a particular level of work in process (WIP) depend upon costs for higher levels of WIP, making it impossible to formulate a recursive solution. For each possible configuration of intermediate inventories a production policy must specify at which stage to produce next and the number of units to be processed. We prove that any arbitrarily “fixed” production policy gives rise to a finite set of linear equations, and develop algorithms to solve the two-stage problem. We also show how the general 2-BNS can be reduced to a three-stage problem, where the middle stage is a non-BN, and that the algorithms developed can be modified to solve this problem.

Manufacturing control of a serial system with binomial yields, multiple production runs, and non-rigid demand: a decomposition approach

In this paper, the problem of ordering a series of production runs to meet a single make-to-order demand, where the various stages in the production process have binomial yields, is addressed. The use of additional procurement of part-finished material or reworking defective material previously made to supplement yield at any stage is considered. The costs of this and manufacture, disposal of surplus or shortage are assumed to be proportional to the numbers involved. Set-up costs may or may not be incurred. The treatment follows from, and builds on, a previous single-run analytical result by developing an approximation in which the problem is decomposed into a series of single runs.

The multiple lot sizing problem with rigid demand and interrupted geometric yield

We study the multiple lot sizing problem with interrupted geometric yield distribution. Compared to the binomial and discrete uniform yields, the lot sizing problem with interrupted geometric distribution is less studied. We present results characterizing the behavior of the cost function and optimal lot sizes. These results help us understand the imperfect processes described by the interrupted geometric distribution. We also develop bounds which lead to an efficient numerical procedure with an approximately linear computation time for large problems.

Multiple lot-sizing, and value of probabilistic information, in production to order of an uncertain size

A make-to-order batch manufacturer knows that an order for items with some particular specifications is forthcoming, but does not yet know the exact size of the order. Due to production lead times, work needs to start immediately. But some units produced may fail to meet required tolerances; that is, the yield of each batch is random. Each new run involves a costly setup, and each unit attempted involves material and other variable costs. If a subjective probability distribution for the order size can be formed, the following related questions arise: 1. (a) What is the optimal run size? 2. (b) When should production stop? To answer these questions, this paper formulates a new model, whose main new feature is the ability to handle uncertain demand within a multiple lot-sizing setting. It is proved that the optimal policy for this scenario is one of control limit — stop if and only if the stock of good units is larger than some critical value. A computer program was developed for solving the problem for binomial yields, and optimal policies for several examples are reported. Due to the effort required to form and communicate probabilistic forecasts, a common practice is to express a forecast as a single number, corresponding, perhaps, to the mean or median of the unarticulated distribution. We compute the extra profit expected from forming, and using, probabilistic forecasts. We also show how to compute the profit's variance for any policy.

The multiple lot sizing problem with rigid demand and interrupted geometric yield

We study the multiple lot sizing problem with interrupted geometric yield distribution. Compared to the binomial and discrete uniform yields, the lot sizing problem with interrupted geometric distribution is less studied. We present results characterizing the behavior of the cost function and optimal lot sizes. These results help us understand the imperfect processes described by the interrupted geometric distribution. We also develop bounds which lead to an efficient numerical procedure with an approximately linear computation time for large problems.

Control Limits for Multi-stage Manufacturing Processes with Binomial Yield (Single and Multiple Production Runs)

The problem of ordering a single or a series of production runs to meet a single make-to-order demand, where the various stages in the production process have binomial yields, is addressed. Use of additional procurement of part-finished products or reworking defective material previously made to supplement yields at any stage is considered. The costs of this and manufacture, disposal of surplus or shortages are assumed to be proportional to the numbers involved. It is shown that when a single production run is considered, the optimal policy is defined by two critical numbers (control limits) at each stage. The treatment of ordering a series of production runs follows from and builds on the single run analytical result by developing an approximation in which the problem is decomposed into a series of single runs. Set-up costs may or may not be incurred.

Single-machine lot-sizing with uniform yields and rigid demands: robustness of the optimal solution

We consider the lot-sizing discrete problem of a single machine whose yield is uniformly distributed. We assume that the customers' demand is rigid, i.e. all demand must be satisfied. The costs involved are a setup cost paid each time a run is initiated plus a unit variable cost per unit produced. No salvage cost is associated with extra or defective units. We prove that there exists an optimal sequence of lot-sizes that is strictly increasing in the demand levels and therefore an optimal lot-size that is at least as large as the demand level. These properties have been proved in the literature only for binomial yields. We also provide an extremely simple algorithm to compute the optimal lot-sizes. Moreover, we show that the cost function is strongly robust in the lot-size: for any ϵ > 0 we develop a procedure that generates a (usually large) class of policies whose relative error is bounded by ϵ.

Single bottleneck systems with proportional expected yields and rigid demand

This study considers single bottleneck systems: multistage production systems where all setup costs except one are zero. The stage with non-zero setup cost is defined to be the bottleneck (BN). Production is in lots and demand needs to be satisfied in its entirety. The expected yield (number of good units) at each stage is proportional to the lot size and only good items may proceed from one stage to another. We reveal several general features of the optimal policy and discuss some specific yields in detail. In particular it is shown that if units are processed one-by-one on the non-BNs, the optimal lotsizing problem can be reduced to the problem of optimally lotsizing a single-stage ‘equivalent machine’. Then, if the BN has Binomial yeild, the ‘equivalent machine’ itself has Binomial yields. Also, examples are given which demonstrate the non-intuitive structure of the optimal policy.

Distributions of Amenorrhoea and Anovulation

Most statistics of postpartum amenorrhea are based on retrospective rather than prospective reporting and except when short (averaging less than 6 months) exhibit bimodality negative skewness and gross heaping on multiples of 6 months. The 2 best prospective series 1 representing moderately long and the other very long postpartum amenorrhea exhibit modest degrees of bimodality and negative skewness. Bimodality is plausibly explained in terms of sample heterogeneity that is a combination of lengths of amenorrhea of successful lactators which might be expected to be unimodal with the brief amenorrhea of nonlactators and of mother experiencing early infant deaths and sometimes of respondents experiencing pregnancy losses or confusing postpartum bleeding for genuine onset of menstruation. To cope with the problems of bimodality and heaping on multiples of 6 months Lesthaeghe and Page have used an ingenious logit regression amenorrhea approach. Logit-fitted curves exhibit positive skewness when the retrospective amenorrhea series is short or very long and negative skewness when the series is of intermediate length. For retrospective series averaging between 6 and 9 months the negative skewness is especially marked with a 2nd mode several months later than the mean. A key issue is whether this feature reflects a real phenomenon related to extreme sample heterogeneity or is an artifact of partially surpressed heaping. Hitherto in the construction of family building models postpartum anovulation has been conventionally represented by Barretts modified Pascal distribution. It is shown than when amenorrhea series are long averaging over a year this distribution generates excessive variances. A generalization of Barretts curve the negative binomial yields control over the variance as well as the mean but also unimodal positively skewed curves of amenorrhea that grossly underestimate the left tail of the distribution. A mixed geometric-negative binomial distribution is proposed which ensures bimodality avoids the deficit of short length and produces a much more moderate degree of positive skewness for series of intermediate length. The present analysis makes clear that considerably more prospective data are needed to prove decisively that the degree of bimodality depends primarily on sample heterogeneity; or to decide whether the strong negative skewness found in some logit-fitted curves is artifactual or real. Correspondingly more prospective data including information about the timing of infant deaths are a prerequisite to exploring how well mixed distributions can simulate bimodal distributions of amenorrhea. (Authors)